Problem 92 - Elementary Algebra

Question

If the sides of a square measure 2 units, then find the length of the diagonal.

Step-by-Step Solution

Answer

The diagonal is \(2\sqrt{2}\) units.

Step 1: Understand the Problem

We need to find the diagonal of a square when the sides are 2 units each. A square has four equal sides, so each side in this case measures 2 units.

Step 2: Recall the Diagonal Formula of a Square

The formula for the diagonal of a square with side length \(a\) is given by \(d = a\sqrt{2}\). This comes from the Pythagorean theorem where a diagonal divides the square into two right-angled triangles.

Step 3: Substitute the Value of the Side into the Formula

Substitute \(a = 2\) into the diagonal formula: \(d = 2\sqrt{2}\).

Step 4: Calculate the Numerical Value

Calculate \(2\sqrt{2}\). Approximate \(\sqrt{2} \approx 1.414\), thus \(2\times1.414 = 2.828\). However, the exact value is \(2\sqrt{2}\) since we typically leave it in this form unless otherwise specified.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that explains the relationship between the sides of a right triangle. It states that for any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is expressed mathematically as: \[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides. In the case of the square, the diagonal acts as the hypotenuse of two right-angled triangles formed where the diagonal crosses. Each side of the square becomes one of the triangle's shorter sides. This relationship allows us to use the sides of the square to calculate its diagonal, ensuring precise measurements based on these geometric principles.

Square Properties

A square is a special type of quadrilateral where all sides have equal length and each angle measures 90 degrees. This regularity in shape gives squares unique properties that make them particularly easy to work with in geometric calculations. Here are some key properties:

• All sides are equal.
• Opposite sides are parallel.
• Diagonals bisect each other at right angles, dividing the square into two equal right triangles.
• Diagonals are equal in length.

The properties of a square are crucial for understanding how its diagonal is calculated. Because the diagonals are equal and bisect each other at right angles, they can serve as the hypotenuse for applying the Pythagorean Theorem, providing a straightforward method to find the diagonal length when the side length is known. Squares maintain their symmetry regardless of size, which simplifies many mathematical processes, including this kind of diagonal computation.

Algebraic Calculation

Algebraic calculation allows us to solve geometric problems such as finding the length of a diagonal using formulas. With a square, the formula for the diagonal is \[d = a\sqrt{2}\] where \(d\) is the diagonal and \(a\) is the length of a side. This formula is derived from the Pythagorean Theorem and makes calculating the diagonal straightforward once you know the length of a side. To apply this formula:

• Identify the length of one side of the square. For instance, in this exercise, each side is 2 units long.
• Substitute the side length into the formula: \(d = 2\sqrt{2}\).
• Simplify the expression as needed. Here, the simplified form remains \(2\sqrt{2}\).

While it is often left as an exact expression \(2\sqrt{2}\), an approximate decimal can also be calculated if needed, such as \(2.828\). This method brings simplicity and precision to solving problems involving squares.